While randomization has long been used profitably in
optimization and computer science, it's importance in algebraic
geometry has only recently been realized. In particular, the
study of random systems of equations leads to some beautiful
interactions between geometry, applications, and algorithmic complexity.
We survey some of these developments from the point of view
of polynomial system solving, focussing particularly on real
(as opposed to complex) solutions. We will also see how centuries-old
questions on counting real solutions can be solved with modern ideas
from algebraic geometry (and a little randomization).
No background in algebraic geometry or algorithms is assumed.
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Last modified: April 11 2016 - 18:14:43